https://www.johndcook.com/blog/2019/06/20/bounds-on-the-nth-... is a recent blog posted titled "Bounds on the nth prime". It lists tighter upper and lower bounds:

    from math import log
    def f(n, k):
      ln = log(n); lln = log(log(n))
      return ln + lln - 1 + (lln-2)/ln - ((lln**2) - 6*lln+k)/(2*ln*ln)

With that in place, and once I realized that 2^1017 mean 2E17 not 2 to the power of 1017:

   >>> n = 2E17
   >>> lo = n * f(n, 11.847)
   >>> hi = n * f(n, 10.273)
   >>> lo
   8.512627944213742e+18
   >>> hi
   8.512727125430618e+18
   >>> exact = 8512677386048191063
   >>> (exact - lo)/exact
   5.80802398678381e-06
   >>> (exact - hi)/exact
   -5.842977499434444e-06